In this talk, after a general outline of the topic, we introduce the framework of semi-abelian categories and our main tool, namely basic functor calculus. It gives rise to the construction and basic properties of a new concept of categorically defined commutators which in the semi-abelian context encompasses earlier notions in the literature and renders them much easier to handle. In particular, we deduce the notions of upper and lower central series and of derived series which generalize certain classical or recent constructions in specific contexts.